The Damped Nonlinear Pendulum

This demonstration illustrates a simple
phase plane plot. The particular system plotted in this example is

x'=y
y'=-sin(x) -0.1 y

which is equivalent to the standard equation of motion

x"+0.1x'+sin(x) = 0

for the damped nonlinear pendulum.
In this application, the variable x is really the angle theta between the
pendulum arm and vertical and y is x', which is proportional to the speed
of the pendulum bob. The animated pendulum shows the state of the pendulum corresponding to the current point of the trajectory in the phase plane plotter.

To plot the trajectory that starts at any specific point, just click
the mouse there. The program stops plotting a trajectory when it
reaches the edge of the screen or when the user clicks the "Stop"
button. To restart a stopped trajectory click "Continue".
The user may clear the screen by clicking the "Clear" button.
By default, trajectories are plotted forwards in time.
The user may also chose to have trajectories plotted backwards in time.
Just click the "Forwards" button. It toggles between "Forwards" and
"Backwards".